Asymmetric relation

In mathematics an asymmetric relation is a binary relation on a set X where:

• For all a and b in X, if a is related to b, then b is not related to a.

In mathematical notation, this is:

$\forall a,b\in X,\ aRb\;\Rightarrow \lnot (bRa)$ .

Examples

An example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarski's axioms characterizing the real numbers R is that < over R is asymmetric.

An asymmetric relation need not be total. For example, strict subset or ⊊ is asymmetric, and neither of the sets {1,2} and {3,4} is a strict subset of the other. In general, every strict partial order is asymmetric, and conversely, every transitive asymmetric relation is a strict partial order.

Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation is the rock-paper-scissors relation: if X beats Y, then Y does not beat X, but no one choice wins all the time.

The ≤ (less than or equal) operator, on the other hand, is not asymmetric, because reversing x ≤ x produces x ≤ x and both are true. In general, any relation in which x R x holds for some x (that is, which is not irreflexive) is also not asymmetric.

Asymmetric is not the same thing as "not symmetric": a relation can be neither symmetric nor asymmetric, such as ≤, or can be both, only in the case of the empty relation (vacuously).

Properties

• A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
• Restrictions and inverses of asymmetric relations are also asymmetric. For example, the restriction of < from the reals to the integers is still asymmetric, and the inverse > of < is also asymmetric.
• A transitive relation is asymmetric if and only if it is irreflexive: if a R b and b R a, transitivity gives a R a, contradicting irreflexivity.